A technique known as "harmonic balance" is commonly used to provide fast and accurate steady-state analysis of nonlinear circuits. The harmonic balance technique is particularly well-suited for analysis of nonlinear circuits driven by two or more periodic inputs at widely separated frequencies. Unlike other conventional analysis techniques, the computation time required for a harmonic balance analysis is substantially insensitive to the actual numeric values of the stimuli frequencies. The harmonic balance technique generally involves expressing each of n circuit node waveforms as a Fourier series truncated to N coefficients, then replacing the differential equations describing the circuit operation with a system of nonlinear algebraic equations involving the Fourier coefficients. This replacement is possible because the derivative with respect to time of a Fourier series is an algebraic operation. A numerical technique, such as Newton's method, is then employed to solve the resulting system of nonlinear equations. Additional details regarding the harmonic balance technique may be found in, for example, K. S. Kundert, J. K. White and A. Sangiovanni-Vincentelli, "Steady-State Methods for Simulating Analog and Microwave Circuits," Kluwer, Boston, Mass., 1990; R. J. Gilmore and M. B. Steer, "Nonlinear Circuit Analysis Using the Method of Harmonic Balance--A Review of the Art. Part I. Introductory Concepts," Int. J. on Microwave and Millimeter Wave Computer Aided Engineering, Vol. 1, No. 1, 1991; and V. Rizzoli, C. Cecchetti, A. Lipparini and F. Fastri, "General-Purpose Harmonic Balance Analysis of Nonlinear Microwave Circuits Under Multitone Excitation," IEEE Trans. Microwave Theory and Tech., Vol. MTT-36, pp. 1650-1660, December 1988, all of which are incorporated by reference herein.
As noted above, n is used to denote the number of node waveforms which must be stored to capture the behavior of the circuit being analyzed. Even a medium-sized integrated circuit (IC) can have hundreds of nodes, especially if a sophisticated transistor model is used. The transform dimension N can also become large in the case of a multi-tone analysis. For the simulation of a mixer, for example, a designer might request 16 harmonics of the mixer local oscillator and 8 harmonics of the input signal frequency. Because two real numbers are needed to capture the amplitude and phase of each spectral component, the final transform dimension is on the order of N=2.times.8.times.16=256. The overall dimension of the system of harmonic balance equations is nN, which can become quite large in practical applications. Letting X denote the aggregate vector of all unknown Fourier coefficients, the harmonic balance equations may be written as a harmonic balance system H(X)=0 of size nN.
The preferred conventional methods for solving harmonic balance equations are Newton-based techniques due to their fast convergence and high accuracy. Such techniques are described in, for example, J. M. Ortega and W. C. Rheinboldt, "Iterative Solutions of Non-linear Equations in Several Variables," Academic Press, New York, 1969, which is incorporated by reference herein. However, most implementations of Newton-based techniques require the use of a Jacobian matrix .differential.H/.differential.X which is of dimension nN and is generally much more dense than conventional matrices which arise in numerical circuit simulation. As a result, inversion of the harmonic balance Jacobian matrix becomes a computational bottleneck and is largely responsible for the limitation of conventional harmonic balance techniques to analysis of relatively small circuits. Other techniques have been proposed which attempt to simplify or avoid the inversion of the Jacobian matrix. For example, V. Rizzoli, F. Mastri, F. Sgallari and V. Frontini, "The Exploitation of Sparse Matrix Techniques in Conjunction with the Piece-Wise Harmonic Balance Method for Nonlinear Microwave Circuit Analysis," IEEE MTT-S Int. Microwave Symp. Digest, Dallas, pp. 1295-1298, May 1990, which is incorporated by reference herein, attempts to solve the harmonic balance equations using approximate inversions of the Jacobian matrix. A technique described in S. A. Maas, "Nonlinear Microwave Circuits," Artech House, 1988, which is incorporated by reference herein, attempts to solve the harmonic balance equations without using the Jacobian matrix at all.
However, there are a number of advantages associated with exact inversion of the full harmonic balance Jacobian matrix. First, Newton's method with accurate Jacobian matrix inversion exhibits local quadratic convergence, that is, a solution to the harmonic balance system can be obtained to machine precision in a fixed, small number of iterations. Thus, the overall solution time would be small if the accurate Jacobian inversion could be done efficiently. Second, widely convergent continuation methods for solving the harmonic balance equations generally also require solving linear systems involving the Jacobian matrix as a sub-matrix. These widely convergent continuation methods are described in greater detail in, for example, L. Watson, "Globally Convergent Homotopy Methods: A Tutorial," Appl. Math. and Comp., Vol. 31BK, pp. 369-396, 1989, which is incorporated by reference herein. Finally, the inverted Jacobian matrix can be interpreted as a linearization of a circuit in a small-signal sense, and can therefore be utilized for sensitivity computation and noise analysis, as described in S. W. Director and R. A. Rohrer, "Automated Network Design--the Frequency Domain Case," IEEE Trans. Circuit Theory, Vol. CT-16, pp. 330-337, 1969, and R. A. Rohrer, L. Nagel, R. Meyer and L. Weber, "Computationally-Efficient Electronic Circuit Noise Calculations," IEEE J. Solid State Circuits, Vol. SC-6, pp. 204-213, 1971, respectively, both of which are incorporated by reference herein. Additional support for the need for accurate Jacobian inversion may be found in V. Rizzoli, F. Mastri and D. Masotti, "A Hierarchical Harmonic Balance Technique for the Efficient Simulation of Large Size Microwave Circuits," 25.sup.th European Microwave Conference, Bologna, Italy, September 1995, pp. 615-619, which is incorporated by reference herein. Fast and accurate inversion of the Jacobian matrix is therefore a key enabling technology for further progress in the analysis of large nonlinear circuits as well as in other important applications of the harmonic balance technique.
It is apparent from the above that a need exists for improved techniques for inverting or otherwise processing a harmonic balance matrix generated as part of a frequency domain analysis of a large nonlinear circuit or other device, such that the storage and computation requirements associated with processing the matrix are considerably reduced.